document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1143646>Question 1143646</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The equation of the plane passing thru points (5,4,1) ,(4,-2,-3) and (0,6,5) is expressed as x/A+y/B+z/C=1  <BR>\n" );
document.write( "Find the value of B<BR>\n" );
document.write( "Find the value of C<BR>\n" );
document.write( "Find the equation of the plane. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #764545 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=764545\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The equation of the plane passing thru points Q(5,4,1), R(4,-2,-3) and S(0,6,5) is expressed as x/A+y/B+z/C=1  <BR>\n" );
document.write( "Find the value of B<BR>\n" );
document.write( "Find the value of C<BR>\n" );
document.write( "Find the equation of the plane.<BR>\n" );
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document.write( "Step 1, define 2 vectors, QR and RS.<BR>\n" );
document.write( "Q - R = (1,6,4)<BR>\n" );
document.write( "QR = i + 6j + 4k<BR>\n" );
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document.write( "R - S = (4,-8,-8)<BR>\n" );
document.write( "RS = 4i - 8j - 8k<BR>\n" );
document.write( "----<BR>\n" );
document.write( "Step 2 find the cross product of QR and RS<BR>\n" );
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document.write( "|i   j   k|\r\n" );
document.write( "|1   6   4| = i*(-48+32) - j*(-8-16) + k*(-8-24) = -16i + 24j - 32k\r\n" );
document.write( "|4  -8  -8|\r\n" );
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document.write( "</PRE><BR>\n" );
document.write( "This the vector normal to the plane.<BR>\n" );
document.write( "Since we're not concerned with its magnitude, reduce it to<BR>\n" );
document.write( "2i - 3j + 4k\r<BR>\n" );
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document.write( "Call it N.<BR>\n" );
document.write( "--------------------<BR>\n" );
document.write( "Derive the equation of the plane.<BR>\n" );
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document.write( "The equation of the plane is Ni(x-a1) + Nj(y-a2) + Nk(z-a3) = 0, where (a1,a2,a3) is any point in the plane and (Ni,Nj,Nk) is the normal vector, N.\r<BR>\n" );
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document.write( "In the example, using point Q, which is (5,4,1), the equation of the plane is 2(x-5) - 3(y-4) + 4(z-1) = 0, which simplifies to\r<BR>\n" );
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document.write( "2x-10 - 3y+12 + 4z-4 = 0, or <BR>\n" );
document.write( "2x - 3y + 4z = 2\r<BR>\n" );
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document.write( "Verify your answer. Substitute the original points to see if they satisfy the equation of the plane. <BR>\n" );
document.write( "(5,4,1)<BR>\n" );
document.write( "2*5 - 3*4 + 4 = 2 OK<BR>\n" );
document.write( "(4,-2-3)<BR>\n" );
document.write( "2*4 + -3*-2 + 4*-3 = 2 OK<BR>\n" );
document.write( "(0,6,5)<BR>\n" );
document.write( "2*0 - 3*6 + 4*5 = 2 OK<BR>\n" );
document.write( "The 3 points are in the plane.<BR>\n" );
document.write( "=====================================<BR>\n" );
document.write( "To express it in the form x/A+y/B+z/C=1 divide by 2.<BR>\n" );
document.write( "x/2 + y/(-2/3) + z/(1/2) = 1<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "Find the value of B<BR>\n" );
document.write( "B = -2/3<BR>\n" );
document.write( "Find the value of C<BR>\n" );
document.write( "c = 1/2<BR>\n" );
document.write( "=============================<BR>\n" );
document.write( "PS  I made an Excel sheet that does these, if you're interested. </SPAN>\n" );
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